To determine the number of real solutions of the quadratic equation \(4x^2 - 3x + 2 = 0\), we first need to identify the coefficients of the quadratic equation in the standard form \(ax^2 + bx + c = 0\).
For the given equation:
- \(a = 4\)
- \(b = -3\)
- \(c = 2\)
The discriminant \(D\) of a quadratic equation is given by the formula:
\[ D = b^2 - 4ac \]
Now we can substitute the values of \(a\), \(b\), and \(c\) into the formula:
\[ D = (-3)^2 - 4(4)(2) \]
Calculating \( (-3)^2 \):
\[ D = 9 - 4(4)(2) \]
Calculating \( 4(4)(2) \):
\[ 4(4)(2) = 32 \]
Now, substituting this back into the equation for \(D\):
\[ D = 9 - 32 \]
Computing \(9 - 32\):
\[ D = -23 \]
Part 1: Value of the Discriminant
The value of the discriminant \(D\) is \(-23\).
Part 2: Determine the Number of Real Solutions
The discriminant tells us about the nature of the roots:
- If \(D > 0\): Two distinct real solutions
- If \(D = 0\): One real solution (a repeated root)
- If \(D < 0\): No real solutions (the roots are complex)
Since the discriminant \(D = -23\) is less than 0, we can conclude that:
There are no real solutions to the equation \(4x^2 - 3x + 2 = 0\).
1 answer